How can I see which registry keys are written during the installation of a program?

Use Process Monitor for this purpose. RegMon and FileMon are merged into this and it can now be used to monitor process, registry and files.

Revo Uninstaller will track an installation and show you all files and registry entries that are written during an installation.

Use a software that does Registry Snapshots like RegShot2.

Regshot2 Unicode is a registry change and file system change detection diff tool that takes snapshots, generates HTML reports, and creates automatic REG undo/redo scripts. The program can save/load snapshots to file for later use, while registry path and file path inclusion/exclusion rules can be set to include/exclude paths in the snapshots.

• Do a registry snapshot
• install the software
• do a 2nd snapshot and let the software compare the 2 snapshots for diffs.

Now you see the new/altered registry settings.

The best tool for this is process explorer by Windows Sysinternals. It's free tools that can view file and registry handles of any program running on the system.
Here's the link: https://docs.microsoft.com/en-us/sysinternals/downloads/process-utilities)

Here's an image of the open handles of Explorer.exe